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Second-Order Logic, Or: How I Learned to Stop Worrying and Love the Incompleteness Theorems

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Second-Order Logic, Or: How I Learned to Stop Worrying and Love the Incompleteness Theorems SECOND-ORDER LOGIC, OR: HOW I LEARNED TO STOP WORRYING AND LOVE THE INCOMPLETENESS THEOREMS ETHAN JERZAK Abstract. First-order logic enjoys a nearly universal status as the language in which mathematics is done|or at least, in which it is ideally done. Whence this priority? In this paper, I present some of the basic principals of first- order logic, and then give an introduction to second-order theories. I shall demonstrate that second-order logic is not reducible to first-order logic, and I will sketch some of the meta-logical results that make first-order logic so appealing and that fail for higher orders. The emphasis shall be more on explication than on detailed proof: my aim is to sort out what one gains and loses by limiting himself to first-order quantifiers, and to see what second-order theories can offer in terms of expressive power. Contents 1. History 1 2. First-order Logic 3 2.1. Syntax 3 2.2. Semantics 4 2.3. Deductive Systems 5 3. First-Order Metalogical Results 5 4. First-Order Limitations 8 4.1. Skolem's Paradox 8 4.2. `A Paradoxical Situation' 9 4.3. Ultraproducts 9 5. Second-Order Logic 12 5.1. Deductive Systems 13 5.2. Semantics 14 6. Second-Order Metaresults 15 7. Conclusion: Living without Completeness 18 Acknowledgments 21 References 22 1. History These historical considerations may well seem extraneous to the contemporary mathematician|or, if not exactly extraneous, at least as a skimmable first section written merely to orient oneself vis-a-vis certain incidental timelines. It is, as one says, of merely historical interest. But nonetheless I proceed with it, resolutely Date: 21 August 2009. 1 2 ETHAN JERZAK ignoring the consequent jeers and slanders of the ahistoricists. This is not a paper on mathematics; it is a paper on the foundations of mathematics, and as such historical considerations are of more than merely cursory interest. Progress in mathematics as such is basically unidirectional, as long as one does not forget the fundamental theorems. Progress in the foundations of mathematics, on the contrary, is sketchy and multifarious, precisely because there is no definitive notion of deduction for these more philosophical questions. We may know more mathematics than Kant, but this does not exclude the possibility that his theory of its foundations does not remain more correct. To history, then, without further apology. Mathematics today conforms to a self-conception that is relatively new (certainly fewer than 150 years old). Roughly, the contemporary mathematician lays down axioms and definitions more or less arbitrarily, and then works out the consequences of these axioms according to a deductive system of some sort. No longer does the mathematician hold that his axioms are in some way the `correct' axioms for de- scribing the world; the analyst remains neutral on whether the universe really looks like R3 or not. The focus instead is on giving deductive proofs from these axioms and definitions. Some conventional shorthand is generally allowed|to write out even fairly basic proofs fully would be unacceptably time-consuming|but gener- ally one considers his proof valid if any competent person could easily translate it into a valid argument in some first-order language, had he only the time and moti- vation. (Of course, the level of dogmatism on this point varies between the different branches, but most of the mainstream divisions embody it to some degree.) This notion of mathematical proof differs greatly from its historical predecessors. Most proofs prior to Frege were written in something very close to plain English (or Greek, German, or French, as the case may have been). Kant (1724{1804) thought that a mathematical proof was not a purely analytic thing; the pure imagination was involved in determining constructibility, such that going through a proof was a genuinely active process that only a rational agent with all of the cognitive faculties could undertake. The notion of a purely formal language developed first with Gottlob Frege (1848{1925), whose Begriffschrift was supposed to bridge `gaps' in the natural language proofs and resolve certain contradictions that ambiguities in natural languages allowed. For example, the Intermediate Value Theorem was taken to require pure intuition, not logical proof, until Bolzano supplied an analytic proof from the definition of continuity in 1817. Frege wanted to similarly ground all of mathematics.1 This project spawned logicism, the idea that mathematics could be reduced in its essentials to logic (and thereby reducing mathematics to an analytic, not synthetic, discipline). It is here that logic began to take the form that we know it today, and, indeed, where the notions of first-order logic became separated from higher orders. Logic emerged as a purely formal deductive system that could be studied in its own right, and upon which all mathematics was supposed to be constructed. The development of formal logic, then, is historically bound up with a certain brand of foundationalism. One must find firm ground on which the structures of mathematics can be secured, and the way one does it is generally via what became today's first-order logic. I shall give a brief overview of bread-and-butter first-order 1One notable exception is Geometry. Frege actually agreed with Kant that Geometric proofs were in fact synthetic, and required pure intuition in addition to logical deduction. SECOND-ORDER LOGIC, OR: HOW I LEARNED TO STOP WORRYING AND LOVE THE INCOMPLETENESS THEOREMS 3 logic, in order to then give a slightly more detailed overview of second-order logic and compare the foundational merit of each. 2. First-order Logic In its broadest sense, we take `logic' to mean `the study of correct reasoning'. We want to formalize such intuitive notions as `valid arguments' and `proofs'. To accomplish this goal, we will develop a formal system into which we can translate what we take to be correct arguments in English and ensure that they are indeed complete and correct. Ideally, we will develop a formal language such that an English argument is valid if and only if it is modeled by a formally valid inference in our language. Needless to say, this is a goal perhaps too ambitious for mere mortals. Thus, we shall restrict ourselves to the language commonly spoken by mathematicians. The important thing is always to keep in mind is that logic is a model for correct reasoning whose justification stems, at least initially, from our plain-English intuitions of what constitutes a correct argument. We seek a language that captures this intuition as faithfully as possible. One may, if one wishes, be on guard against a sort of epistemological circle here: we use intuitions to develop and justify a system of correct reasoning, and in turn make determinations about the correctness of our English-language arguments based on whether it is successfully modeled in that formal language. This worry I leave aside for now; let us at least explore the circle before trying to find a way out of it. first-order logic consists of two main parts: the syntax, and the semantics. Roughly speaking, the syntax formally determines which strings of symbols are permissible for the language; the semantics assigns meanings to these permissible expressions. As we will see, in standard first-order logic, these two notions will coincide nicely. The exposition of first-order logic that follows is a bit hasty, and serves mainly to establish the notation that will be drawn upon in later sections. 2.1. Syntax. The syntax of a language consists of an alphabet together with for- mation rules. For first-order logic, we have an alphabet consisting of logical symbols that retain the same meaning no matter the application: • Quantifiers: 8 (universal) and 9 (existential) • Logical connectives: : (negation), ^ (conjunction), _ (disjunction), ! (conditional), and $ (bicontitional) • Punctuation: ( and ) • A set of variables: fx0; x1; x2:::g • Identity: = (optional) And we also have some non-logical symbols, depending on the application. Non- logical symbols represent relations, predicates, functions, and constants on whatever we take to be our domain of discourse. For example, if we were doing group theory, we would want the symbols f1; ∗} always to function in the same way between different groups. If we were doing arithmetic, we would want f0; 1; +; ∗; <g at our disposal. In general, we allow the set of non-logical symbols to be as large as one 4 ETHAN JERZAK pleases. It can be any cardinality.2 Call a first-order language with a set K of non-logical symbols L1K. If it has equality, call it L1K =. A set of symbols alone is insufficient for making a meaningful language; we also need to know how we can put those symbols together. Just as we cannot say in English \Water kill John notorious ponder," we want to rule out pseudo-formulas like \8(! x12_." We therefore define a well-formed formula by formation rules. I omit the details, which are tedious and can be found in any competent text on first-order logic. One first defines the terms inductively as any variable xn or any expression f(t1:::tn) where f is an n-place function and ti are terms. One then specifies certain formulas as atomic: usually those of the form P (t1:::tn) where P is any n-place relation and t1:::tn are terms. One then wants to make provisions for building complex formulas out of the atomic ones, through rules such as: if φ and are formulas, then :φ, φ ! , and (8xi)φ(xi) are all formulas. These three rules are actually sufficient to generate all the formulas, once one trims away the excess connectives and quantifiers.
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